Let's say you have a sphere-maze.
The sphere is fixed on the fixed half-ring (1) and two independent rings (2) and (3). These rings are connected at point A. Inside the sphere one can see a small ball (point B). When you are rotate two rings, the ball begins to spin.
Suppose that the initial position of the sphere is the combination of a half ring (3), two rings (1, 2) and a red great circle (point G in the photo) on one plate.
On the sphere, the sticker is placed. The player must rotate the sphere so as to place the ball on the sticker. The angle between the rings (2) and (3) is a hint for next puzzle.
Question. How to check (prove) that the puzzle has an unique solution?